One of my holiday plans is to read a bit of Badiou. I’ve got a knee jerk reaction anti, my Lenin allergy, but people I like and respect use him so I’m trying my best to maintain an open mind. That’s not always my forte, and even more so w/ Badiou - all that math! It’s even harder because my mom’s a math teacher and though I am too old I am still playing out adolescent rebellion. I read this stuff and feel by turns hostile and out of my league (I suspect the latter is partially a source of the former).
I’ve just read the translators’ intro to Infinite Thought. I like the part about the football team as an analogy for a set - each member of a set itself contains various subsets, and the set may itself be a subset of another set. The term for this is inconsistent multiplicity. I’ll try to remember that. I’m not convinced of the need for math to make this point, though. One could also just insist on a general infinitude of predicability and of idioms within which predication can occur. This will be a general question as I read this - is the claim that set theory is a useful tool for some people in some circumstances, or is it an attempt to position set theory as a better tool than others? (IE, is Badiou just into math, or is he trying to convince me I ought to be?)
“Precisely because a situation provokes the question ‘What was there before all situations?’ but provides no possible access to this ‘before’ that is not irremediably compromised by post-situational terminology and operations, it is impossible to speak of in any direct way.” (13) Perhaps. Or this before might simply express an ability to speak in ways that don’t make sense. (A la I believe Augustine, or was it Aquinas?, who asked ‘if God made time, when did he do it?’. Time being the condition for ‘when’, the question can’t be answered because any when already implies time. There’s nothing here to be spoken of in some implied indirect way. It’s just not a fruitful question, except for perhaps some other rhetorical effect or koan use or something.)
Mathematics is ontology. What? As in “math is really about being” or as in “ontological talk is really about math”? Ah, “Set theory is the formal theory of non-unified multiplicities. (14)” Another way to rephrase my question above is quite simply, what’s the use of formal languages? Is Badiou using formalization for some purposes, or is he recommending that formalization is a better avenue than others such that I should take it up?
I like the point about multiplicities that are multiplicities vs apparent multiplicities that are really identities, that’s the criticism he makes of Deleuze I think, that D does the latter but thinks he does the former. But I suspect there’s an implied claim lurking which will come out later, that doing so has some sort of political consequences. I’m always torn on this, between on the one hand wanting a radical heterogeneity of discursive registers (because people bad ideas don’t automatically mean doing bad things: first, it simply is the case that people are capable of doing good things with bad ideas, and second this implies people are simply propositional machines carrying out the outcomes of their ideas), and on the other hand wanting to say that of course ideas do matter, bad ideas are a problem, we should have better ones etc. So when the implied claim comes out that identity (unitary) thought makes for political problems, if it does, I’m not a priori opposed, but I’m not immediately on board either. (Life in suspension, that’s me. Fuck.)
Void: (all of) that which conditions a set and which the set does not contain. But why call this nothing? It’s only nothing from the interior of a given set, but from a set which contains that set the constituting may well be present (the constituting element of the subset, not the of the larger set). Why privilege the intra-set view rather than a meta- perspective of the relationship of set to subset? (Hmm, a whiff of dialetics here, perhaps…) This nothing/void stuff will be another thing to ask questions about as I read. It strikes me that absence in a given set is not the same as nothing and does not authorize the predication of ‘nothing’, but rather only ‘absent in X set’. Every set may have such an internally absent element that constitutes it, but to say that therefore the ultimate foundation is itself negative strikes me as only sensible if one holds to a sameness of discursive registers, that is, if the formalized register of set theory is equal to or superior to other registers within which the absent constitutive element might appear with positive traits.
Null set. What in the hell…? Definitely out of my league. But also… a set with no members? What would that mean? Is this simply a formalist maneuver, or is there some claim about something in the world here? It can’t be the latter, by definition. Such that the set with nothing in it would be categorically different than all other sets made up of something(s).
“set theory ontology is indifferent to the existence or non-existence of particular situations” (23). Yeah? So’s your mother…! But, it does “present the ontological schemas of any ontological claim; that is, it presents the structure of what any situation says exists.” (23) Is this a claim about “says” or about “exists”? That is to say, is this a claim about what it means to make a claim (speakers), or is this a claim about all of that which is subject to having a claim made about it (world)?
Distinction of natural, neutral/normal, and historical situations. The example of a fish pond for the natural, which stays in a condition of stasis. “nothing really changes: barring another natural catastrophe the ecosystem will remain in a state of homeostasis.” (27). Yes of course, barring that. But barring historical change historical situations will remain in a state of homeostasis. Leaving aside issues of, say, introduction of non-native fauna (these could, via a questionable human/other animals distinction, be held as not native but historical), there are still matters of, say, volcanic eruption, earthquakes, etc, which disrupt homeostasis. Not to mention, depending on how one reads the term, natural selection, and mutation.
Event… I don’t get it. An event that changes nothing because no one recognizes its importance? (27) What would that be? Sounds to me more like a potential event which fails to happen… Maybe I misunderstand the use of the term, though.
“Ontology does not discern the nature of any situation” but rather “only speaks about the structure of multiplicity: it has nothing to say about the qualities or identity of any concrete situation.” (32) In which case one can ask, why do it? I do like this, though, mainly as a negative move against the ontological turn in Negri and elsewhere. If ontology says nothing specific then it’s not specifically useful. Rather like X in an equation where every term has an X attached: X(15Y) - 3X. We can factor out the X, get X(15Y - 3), and not address it internal to the situation. I like that as well, but only as a negative move: being doesn’t tell us very much.
“if you want to do politics, go become an activist (…) If you want to do philosophy” you’re welcome to, but “don’t confuse the two.” (33) Hmm. I want to agree, but that seems to glib as well.
Basic question - are all these claims etc claims in/about formal languages, or are they bigger claims, and if the latter then are they epistemological or ontological claims? Particularly the claims about emptiness/void/nothing etc…
I think this is the crux of Schelling’s argument against Hegel in On The History Of Modern Philosophy - Hegel tries in the Science of Logic to say “Being is our starting point, the first and presuppositionless thing. Being, qua being, has no predicates. That means it’s nothing. Nothing is synonymous with being. And therefore nothing qua nothing is being, because it has the same (non)predicates. The one term turns over immediately into the other. And that turning over, that’s becoming…” and the wheels are a-turning. Schelling’s response is simply to say “Being does have predicates. Billions of them, many of which we experience. One only gets to an absence of all predicates by a mental operation of subtracting out all content.” Thus, the idea of a presuppositionless philosophy founders at its beginning, there’s a basic epistemological problem, which says that one can’t go from our position - someone thinking - into the universe itself and claim with certainty that being is this or that… The other objection is that there’s a tautology: “being is…”, “nothing is…” - ‘is’ is a form of ‘to be’, of being. The tautology’s not the problem, though, the denial of being tautological is.
Schelling: “thinking being in general and thinking all being in being” is “merely pretence, since it is an impossibility to think being in general, because there is no being in general, there is no being without a subject (…) objective being is already excluded from the absolutely first thought by its nature, it can (…) only be posited for that to which it is an object; being of this kind can therefore only be the second; from this it follows that the being of the absolutely first thought could only be nonobjective (…) nothing is ever posited except just by the subject.” (139)
So with Badiou as well, is the negative ontology a subjective subtractive procedure, or is it a larger claim to objectivity? I’ll have to keep reading. If time permits I should also try to revisit the Tarski “semantic definition” article, that’s one of the few things I’ve really read about formalization. I like it for its deflationary use, but Rorty can run the argument, I don’t need Tarski for that.
I haven’t read Badiou, so I can’t comment on what he’s saying per se. But from your exposition, while I can’t say he’s misusing or misunderstanding set theory so much as making more of it than it is. I think math raises more ontological questions than it answers because it isn’t about being. What’s it about then? I don’t know. I know the entire edifice can be built from set theory and nothing else, and that includes numbers, i.e., numbers are not primitive objects but can be constructed. What does that mean for ontology? I have no clue.
The null set: a formalism, but a necessary one. For example, there are operations on sets that yield other sets (forgive me if you know this already): the union of two sets (notation ∪) is the set that contains all elements in either set; the intersection of two sets (notation ∩) is the set that contains all elements in common between the two sets. So if we have sets A = {1, 2, 3} and B = {3, 4, 5} then
A ∪ B = {1, 2, 3, 4, 5}A ∩ B = {3}
Now let’s say C = {4, 5, 6}
A ∪ C = {1, 2, 3, 4, 5, 6}BUTA &cap C = {}
Comment by et alia — December 20, 2005 @ 7:44 pm
Woof. I made a nice mess of that. Let’s try again:
A = {1, 2, 3}
B = {3, 4, 5}
C = {4, 5, 6}
A ∪ B = {1, 2, 3, 4, 5}
A ∩ B = {3}
A ∪ C = {1, 2, 3, 4, 5, 6}
A ∩ C = {}
I hope that looks better (is there a way to put a preview button here?) Anyway, the null set is not some claim about the world, but a possible collection—specifically, the one with nothing in it. And yes, it is different from all sets with elements, but that’s nothing to get worked up over.
The null set is the basis of the formal construction of the natural numbers N = {0, 1, 2, 3, 4, 5, …} If numbers aren’t primitive objects how do you build them from sets? Let’s build a set of certain oddball sets. We do this by saying that the null set is in the set and that all the other members can be generated by repeated applications of the formula (I know that “repeated applications of the formula” is vauge and unformalized, but trust me, it can be expressed in terms of sets) S(X) = X ∪ {X} So
S({}) = {{}}
S ({{}}) = {{{}}, {{{}}}}}
S ({{{}}, {{{}}}}} = {{{{}}, {{{}}}}}, {{{{}}, {{{}}}}}}}
I realize that looks like gibberish (and I may have unbalanced brackets there), but the Set N of {} and other sets generated by repeated applications of the rule S(X) = X ∪ {X} has some interesting properties: in particular, certain set opeations and relations on it look exactly like the basic operations and relations of arithmatic. For example, if j and k are both in N then either j ∈ k, j = k, or j ∋ k But it gets better: if j ∈ k then k is the results of some applications of S(j). That means for elements of N ∈ behaves just like < for the natural numbers as we know and love them….
Suddenly, I think I may be causing more harm than good. Email me if this is of any interest. Otherwise, apologies.
Comment by et alia — December 20, 2005 @ 8:11 pm
heya Et,
Thanks for this. I’m not sure I get all of it, but it’s interesting. Can you recommend a decent primer for nonspecialists on this stuff? I haven’t had formal logic or math in long time. While some of this stuff strikes me as obtuse, there’s something about it that’s satisfying as well, and I would like to know this stuff better because some people use it to make points, and I’m always one to collect more means by which to defend my views.
My basic question with all this re: Badiou, particularly the null set and attendant categories (void, nothing), is what this stuff means/is used for outside set theory/math/formal languages. In some sense, I’m not sure I follow what an empty set could mean outside a formal language. As I remember this is one way to describe one of Schelling’s disagreements w/ Hegel, about being and nothing. As I remember it from a course I had in 1999, the argument was presented, basically, as a question of whether being and nothing are predicates like all others or special predicates (the conditions of predication). I think this turned into a question on what one means by nothing, what a thought of nothing might actually be - to my mind, it can’t be, by definition: to say “nothing is…”, by use of the word ‘is’ doesn’t get outside of being and into nothing. The closest there could be would something like “an object which is black and white in all the same places at exactly the same time”, a sort of null set in the sense of one just can’t think it and sort of stops thinking (rather like what I’m told koans are/were supposed to do in zen).
A preview button is a great idea. I’m not very tech savvy. Do you have any idea how I could put one in, or is there a resource you could direct me to to find out? I already pester my computer-knowing friends too much as it is.
happy holidays,
Nate
Comment by Nate — December 20, 2005 @ 8:45 pm
a great intro book on set theory is Mary Tile’s “Philosophy of set theory”… you like to read your philosophy in historical context, G. Frege’s “The foundation of Arithmetic” is absolutely one of the clearest books written on the potential usages of set theory and number theory. The problem is… of course Frege was wrong and abandoned the project himself. Paul Benacerraf has an great “sum up” essay called “On what numbers could not be”… rejecting the set-theory approach to number ontology.
I think Badiou’s interests span this and more, focusing at times on the axioms of set theory itself… esp. on the two of them: the axiom of choice and the continuum hypothesis. Therein lie the formalization of “being” and “event” between the “being” of mathematics and the “event” of construction.
The set theoretic apporach to the question of being and nothing is a unique one in that it doesn’t really bother itself with being a language which conforms with our usual semantic and conceptual difficulties with “being and nothing” and such… rather it forces practioners to speak its language. There are real problems when the null set is excluded as a set. Things either get really complicated or … well… inconsistent. Frege famously used the null set to set the parameters on what the number “1″ means…
The beautiful thing about set theory is that the “sets” aren’t really objects at all, they define an indefinite parameters of possible objects. So, all the things that could ever become associated with the arrangement of “1″ or “73″ or “331″: trees, giraffes, icecream cones, etc. So it rather transforms/rejects “predicate” oriented forms of thinking about so-called “properties.” You might consider the ontology of “possible worlds” as a radical version of this rejection. Thus, in a way, Badiou opens a place to talk about things without “predicates” but rather “names,” with an organization of singularities rather than an organ of interrelated predicate relations. “Formalization” is not a language at all, mathematics, in the tradition of set-theory seeks to get away from the hegemony of discursive units and discouse in general.
Comment by tzuchien — December 21, 2005 @ 7:11 am
An example of an empty set outside of a formal language…OK.
The set of DVDs I own that are movies starring Jim Carrey = {}
The set of books by Ayn Rand that I own = {}
Note that this means the books by Ayn Rand that I own are the same as the movies with Jim Carrey that I own.
I don’t know if the good old empty set can illuminate the philosophical nothing (Heidegger’s Das nicht, is that what he called it?) at all. The empty set is…well, something. It’s a set. What’s a set? A collection of things—most of the time. There are formalizations that go below this level, but I never learned them.
Tzuchien writes: The set theoretic apporach to the question of being and nothing is a unique one in that it doesn’t really bother itself with being a language which conforms with our usual semantic and conceptual difficulties with “being and nothing” and such… rather it forces practioners to speak its language. I’ll buy that, at least the last sentence. It’s hard for me to reflect on what these things might be, but I know what I was taught they could do—at least with respect to each other.
Finally, as regards a preview button, I don’t WordPress, so I couldn’t say. I’m surprised it doesn’t have it, though.
Comment by et alia — December 21, 2005 @ 8:11 am
Thanks to both of you. Interesting stuff. I’ll check out the stuff you recommend as time permits Tzuchien, and have a think on all of this. I’m certainly not trying to make any claims in or against set theory, I don’t know anything about so I can’t do so. My questions reading Badiou are more going to be about the uses of set theory in relation to other stuff. God, it’s been so long since I’ve thought about anything like this… is it Russell, Frege, who… who had the piece on the statement “the present king of France is bald”? Et’s reference of “The set of Ayn Rand novels I own” reminded me of that. Thanks for that example, by the way. One thing that’s interesting and that I think I like in this is a certain generative capacity - one is free to generate sets (for instance, the set of things that are the same as set of Ayn Rand novels owned by Et is infinite).
Tzuchien, can you expand on the point about possible worlds? This whole conversation is about stuff I never understood in the first place when I encountered the little bits of it that I’ve had contact with, and those encounters were several years ago. But it’s all really interesting and fun, so thanks for all the info. I’ll see what I can figure out about a preview button, but it won’t be anytime soon.
take care,
Nate
Comment by Nate — December 21, 2005 @ 9:19 am
Nate: is Badiou just into math, or is he trying to convince me I ought to be?
A little of both, actually. Somewhere in Theoretical Writings he says that all philosophers should stop what they are doing to become mathematicians – though I am sure he doesn’t say this without a bit of a grin and some silent laughter. I am still working with most of this myself , often forgetting, often misreading, but will hazard some input.
You can go to irrational numbers to find pdf’s of Badiou’s Number and Numbers ,where he goes through his own genealogy of set theory, and a transcription of a recent talk given at Birkbeck, where he discusses some of the political importance for applications of Cohen’s generic sets. You are right to point out his use of set theory in the context of his polemic with Deleuze – which is also part of the reason for his insistence that mathematics is an ontology (or rather, simply just is ontology). Badiou considers it a task of any modern philosophy to engage in a thought of multiplicity, which is precisely what Deleuze set out to do. However, Deleuze was to mistake sets as always being numerical, which they are not. Badiou’s own ‘taste’, I suppose, led him via set theory, to reject any form of ‘vitalist’ philosophy or the empirical givenness of sets. It is also in line with his desire to take philosophy and ontology away from ‘approximations’, ‘post-modernism’, and what he terms ‘idealinguistry’. Multiplicity happens to be the only thing that set theory is able to ‘write’ (and there are here some dense Lacanian references for Badiou that at present I do not adequately understand). Set theory therefore deals with being as pure multiplicity, as the question it always answers but never asks. The following I have taken from a letter to Dedekind written by Cantor that appears in From Frege to Godel - definitely not a book to consider as any sort of introduction to set theory and was personally the cause of quite a few headaches:
There is also something crucial missing from your post relating to Badiou’s use of set theory that has to do with how he conceptualizes (and insits on the reality of) truth(s). I don’t want to take up too much space with this so I’ll just skip through some things. The null set (zero, the void, nothingness) is universally included in every set, such that the number three would be written: (0,(0), (0,(0))). Zero and the infinite - neither of which are deducible and remain axiomatic decisions - are synonymous here (at least, that is, in the sense of their having neither inside nor outside). A truth, or event (which does require a subject), reveals that which is the void or infinity of every situation – as that which ‘departs from the set’ thus rendering it ‘momentarily’ inconsitent. It’s that ‘something which was not previously counted in a situation comes to be counted’, i.e. named. I suppose it is something like a ‘potential’ event though Badiou rejects any existence of the virtual. In set theory there is always an excess of how a set can be represented vs. what it presents, which is an actual excess. Badiou always likes to give the example of an amorous encounter: if someone doesn’t notice it, doesn’t name it, then it doesn’t happen and was not an event. This would not, however, prevent it from existing in the situation, as a possibility I suppose. If it is named (the ‘I love you’), then the two faithful subjects go about constructing their world via a ‘truth procedure’ which changes the knowledge of the situation, retroactively. That’s all I can think to write at the moment. Fortunately, the ever-slow translation of Being and Event is near publication, and I suspect a lot of these questions will find answers therein.
Comment by Keith — December 25, 2005 @ 8:42 pm
hi Keith,
Thanks for this. I’ll check out the stuff you recommend. I also plan to take some notes on the next few chapters of Infinite Thought, which I’ve been reading in the interstices of holiday family commitments. Said commitments mean I’ve got less time available for the next few days, but will get back to this when I can. All of this is quite interesting, though I don’t understand a lot of it and don’t know what to do with a lot of it. Generally, I find it quite interesting that Badiou insists on multiplicity, and yet also on universality. These are often held to be incompatible in many encounters I’ve had with them. I don’t know, though. I also don’t get the point about math and ontology. To my mind, the turn to ontology in some (Deleuze, who I don’t really know, and Negri, who I do) is in tension with the multiplicity claims, as they seem to posit a preferred vocabulary while trying to affirm multiplicity. That strikes me as a bit odd, and I’m trying to figure out if Badiou’s doing the same w/ his maths stuff. We shall see. In any case, I like the style he writes in, from what I’ve seen of it thus far. None of the “it both is and not-is, it is its own self-negating unfolding” kind of thing. Very clear presentation, though certainly not easy going.
One question: In a bit of Infinite Thought I read today there’s a reference to Mallarme and the phrase “restrained action”. This same phrase appears in a text that I really love, the manifesto of the Malgre Tout collective, a Franco-Argentine group connected with someone named Miguel Benasayag. Friends of mine translated it and are trying to find a home for it, they translate this phrase as “restricted action”, I’m going to write them to say perhaps “restrained” is a better phrasing. Anyway, does Badiou use this term elsewhere? The term ’situation’ also occurs in a number of excellent contemporary Argentine left-theory works, that’s one of the main reasons I’ve started reading Badiou (that and some other friends of mine).
happy holidays,
Nate
Comment by Nate — December 25, 2005 @ 11:37 pm
Nate -
I was likewise intrigued by Badiou’s insistence on both multiplicity and universality, though I never really considered the terms antithetical. For Badiou, every situation - that is, every multiplicity - is infinite (and he considers this an axiomatic decision)and is therefore, without recourse to the Whole or the One, ‘universal’ in some sense. The math=ontology for some reason always just made sense to me and I wasn’t going to dispute it for any reason. Ontology is the study of what extists, and mathematics is also the study of existence (albeit the existence of number). So, to be a materialist at the same time as a mathematician means what? There are several essays on this in Theoretical Writings.
As far as the “restricted action”, I am not aware of this particular phrase coming up in anything that I have read, though it is very possible that I passed over it without noticing. It does make sense if one were to think about it in relation to Badiou’s “truth procedure”, or his version of a ‘militant engagement’. Every event, in order to be revealed (though nothing is really ‘revealed’) in a situation required the fidelity of a subject, and likewise the careful, calculated, bit-by-bit piecing together of the consequences of a truth. I would understand that to be a sort of “restricted action”, since, as such, it would not be reactionary. It is also important to note that ’situation’ has a very different meaning for Badiou than it would for say, the Situationist.
Happy Holidays as well,
Keith
Comment by Keith — December 26, 2005 @ 4:15 am
hi Keith,
I miss-spoke, I didn’t mean to say there’s an incompatibility b/w universality and multiplicity, just that it’s easy to assume there is. I don’t know enough about all of it to really say. I do find it interesting that there’s a demand on universality in some folks - Badiou and Ranciere - as opposed to an anti-universalism, in the name of left politics.
The stuff on axioms is interesting. Can there be any procedure for determining/producing axioms? Axioms are outside the procedure/sets (sorry if I’m using the terms wrong) they make found/possible. What founds/makes possible axioms? Or are they simply asserted because required? If so, I quite like that - I’m all about tautologies and assertions (much better than infinite regresses). Theoretical Writings is on my list to dive into in the next few weeks, perhaps I’ll have more to say then.
take care,
Nate
Comment by Nate — December 26, 2005 @ 5:04 am
Hey Nate -
I thought Wikipedia had a pretty decent run-down of what an axiom was, as well as the same for axiomatic set theory (Badiou himself uses ZFC). They are asserted because required, I believe. Something like Zermelo’s “axiom of choice” was used and rejected, used, suggested, and used again before it had ever actually been coined by Zermelo. I went to Amazon for the Paul book, which should be at my doorstep by Wed.
- Keith
Comment by Keith — December 27, 2005 @ 2:55 am